In Exercises 33–40, use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=x⁴+5x³+5x²−5x−6;f(3)

Verified step by step guidance

1. Identify the divisor for synthetic division based on the function value you want to find. Since you want to find f(3), the divisor is (x - 3).
2. Set up synthetic division by writing the coefficients of the polynomial f(x) = x^4 + 5x^3 + 5x^2 - 5x - 6. The coefficients are 1, 5, 5, -5, and -6.
3. Perform synthetic division by bringing down the first coefficient (1), then multiply it by 3 (the root from x - 3), and add this product to the next coefficient. Repeat this process for all coefficients.
4. The final number you get after completing synthetic division is the remainder, which equals f(3) according to the Remainder Theorem.
5. Interpret the remainder as the value of the function at x = 3, which is f(3).

Key Concepts

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form (x - c). It simplifies the division process by using only the coefficients of the polynomial, making calculations faster and less error-prone compared to long division.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by (x - c), the remainder is equal to f(c). This means evaluating the polynomial at x = c gives the remainder directly, which is useful for quickly finding function values or checking factors.

Polynomial Evaluation

Polynomial evaluation involves substituting a specific value for the variable x in the polynomial expression and calculating the result. This process is essential for finding function values, verifying roots, or applying the Remainder Theorem efficiently.